Further studies on the composition and microstructure of copolymers

4264

Macromolecules 1992,25, 4264-4280

Further Studies on the Composition and Microstructure of Copolymers by Statistical Modeling of Their Mass Spectra Maurizio S. Montaudo and Giorgio Montaudo' Dipartimento di Scienze Chiniche, Universitb di Catania, and Zstituto per la Chimica e la Tecnologia dei Materiali Polimerici, Consiglio Nazionale delle Ricerche, Viale Andrea Doria, 6, 95125 Catania, Italy Received August 13, 1991; Revised Manuscript Received February 24, 1992

ABSTRACT: Statistical modeling of the mass spectral intensities of copolymers has been used to derive information on the distribution of monomers along the copolymer chain, and an automated procedure to find the composition and the sequence of the copolymers analyzed has been developed. A deconvolution method to determine the microstructure of copolymers when different chemical species contribute to the same mass spectral peak (that is, when a mass spectroscopic peak has an equivocal structural assignment) is presented. The effect of the partial degradation process on the mass spectra of copolymers is discussed, and a theory is given for the interpretation of the mass spectra of copolymers when the cleavage of the copolymer chain occursby a selectiveor nonselectivemechanism. A method is also reported to obtain the copolymer composition by direct analysis of the mass spectra. All these theories have been applied to determine the composition and microstructure of several copolymers (both addition and condensation types) whose mass spectra have been reported in the most recent literature. 1. Introduction

Although polymers often possess molecular dimensions too large to be revealed by high-resolution mass spectrometers (which usually have cut-offs below 10 000 Da), mass spectrometry (MS) has been successfully used to identify the mixture of oligomers formed in the partial degradation of synthetic and biological c~polymersl-~ or of the oligomers contained in low molecular weight polymer^.^,^ We have recently found that decoding of the information contained in the mass spectral intensities leads to the determination of composition and microstructure in copolymers, and this represented significant progress, since only the mass numbers corresponding to MS peaks had been used in previous work on the structural elucidation of copolymers by MS.l*12 Mass spectrometry is a powerful and rapid analytical method and, because it is able to look at the masses of individual molecules in a mixture of homologues, is particularly suitable for the detection of a series of oligomers. However, mass spectra have not been exploited to estimate oligomer distributions, due to the idea192 that a lack of correlation existed between E1 (electron impact) peak intensities and concentration of the oligomers in the mixture. The introduction of soft ionization techniques, such as chemical ionization (CI), fast atom bombardment (FAB), secondary ion mass spectroscopy (SIMS), laser desorption (LD), field desorption and field ionization (FD and FI), etc., has largely eliminated this problem, and much evidence has now accumulated showing that in many cases peak intensities of molecular ions appearing in the mass spectra reflect the relative amounts of oligomers present in the Although this point now appears well established, one should always bear in mind that two major assumptions are being made when correlating MS peak intensities with the actual oligomer concentrations: (a) that the extent of fragmentation of molecular ions deriving from a series of homologous oligomers is not a function of the chain length and (b) that the detection efficiency is the same for all of the oligomers. If that is the case, in order to extract information about the composition and sequence of copolymers, one still has to learn how to exploit MS peak intensities, which depend on the relative abundances of 0024-9297/92/2225-4264$03.0010

the oligomers present in the spectrum and are directly related to the composition and microstructure of copolymers. The statistical analysis of copolymers makes use of Bernoullian and Markovian models to characterize the microstructure of copolymer ~amples.l~-~5 Assuming a theoretical distribution and then fitting the calculated oligomer abundances with the experimental peak intensities (see the example in Figure 11, one can determine the sequence and composition of the In our first approach to the problem,l*l2 we analyzed some microbial copolyesters having Bernoullian distributions, which were subjected to partial methanolysis to reduce their molecular weights. It was assumed that the chain cleavage was a nonselective process. This introduced a limitation in our approach, since the intensity of mass peaks actually depends both on the distribution of comonomers along the chain and on the cleavage process. We found that, from the analysis of the mass spectra, it is possible to ascertain if the cleavage of a copolymer chain occurs by a selective or a nonselective process, therefore removing the above ambiguity, as discussed in detail below. In addition, we have extended our computations to determine the composition and microstructure of several copolymers whose mass spectra have been recently reported by various authors.8J1J621 These spectra have been recorded using different ionization techniques such as LD, FAB, FI, and E1 and refer both to low molecular weight copolymers8J9 (which can be analyzed directly without subjecting the copolymer sample to partial cleavage to reduce their molecular weight) and to high copolymers.11J6-21 The copolymers analyzed are of both the addition and the condensation type, thus removing another limitation in our previous studies, confined only to condensation copo1ymers.'*l2 Finally, one of the great advantages of NMR is that one may immediately get an estimate of the copolymer composition from the ratio of suitable peaks in the NMR spectrum, without performing the statistical calculations necessary to get the sequence information (from which the composition can also be derived).13J4 Mass spectra may provide direct composition estimates as well, and we have now derived simple mathematical formulas that allow us to estimate the copolymer composition, independent of the statistical 0 1992

American Chemical Society

Statistical Modeling of Mass Spectra of Copolymers 4265

Macromolecules, Vol. 25, No. 17, 1992 Table I Number of Peaks Corresponding to Oligomers Expected in the Mass Spectra of a Two-Component Copolymer highest oligomep

peaksb

peaksc

dimers trimers tetramers pentamers hexamers heptamers octamers nonamers decamers 11-mers 12-mers 13-mers 14-mers 15-mers 16-mers

3 7 12 18 25 33 42 52 63 75 88 104 119 135 152

3 6 10 15 21 28 36 45 55 66 78 91 105 120

1

peaksd 6 14 24 36 50 66 84 104 126 150 176 208 238 270 304

4 Highest oligomer group detected in the mass spectrum. * Peaks present in low molecular weight polymers or generated by partiallyselective or nonselective cleavage processes (see section 3). c Peaks generated by totally-selective cleavage processes (see section 3). d Peaks present in low molecular weight polymers or generated by partially-selective or nonselective cleavage processes in the case of two mass series (see section 6).

calculations, by direct analysis of the mass spectra. The latter analysis allows us also to ascertain if the process used to cleave the copolymer chain has been selective or nonselective. Therefore, MS can now be used to provide direct estimates of copolymer composition, analogous to NMR. Statistical modeling of mass spectra of copolymers has thus reached a certain level of generalization, and it can be used to complement NMR a n a l y ~ i s , which ~ ~ J ~has been so far the only technique available for sequencing copolymers.

2. Calculations Mass spectrometry reveals mixtures of oligomers of relatively high molecular weight,'-12 and in our simulation work we have selected examples of mass spectra showing peaks corresponding to oligomers ranging from dimers to 12-, 14-, and 16-mers in some cases. The number of MS peaks corresponding to each oligomer group (dimers, trimers, tetramers, etc.) for an AB copolymer is (n + l), where n represents the number of units. Table I, column 2, reports the total number of MS peaks that can be seen in the mass spectra of AB copolymers. For instance, if the mass spectrum reveals mixtures of oligomers from dimers up to 16-mers, the detection of 152 peaks (all belonging to the same mass number series) might be expected. The high number of MS peaks on which the simulation can be performed defines the microstructure of copolymers with great accuracy. To perform numerical calculations, all the formulas reported later in the text have been incorporated in a computer program, MACO 4, written in Standard ANSI Fortran that runs on a Microvax computer. MACO 4 represents an improved and extended version of MACO ,;lo for a detailed illustration of the program structure we refer the reader to the original description.10 All of the mass spectra of the copolymers that appear in the text have been analyzed employing MACO 4. This work has required a number of improvements both in the theory and in the computer programs (MAC0 3) used previously.10 The present version of the computer program (MACO 4)z2 can deal with all the cases which have appeared in the literature independent of the method by which the partial degradation was performed (methanolysis, aminolysis, photolysis, or pyrolysis).l1J6-21

MACO 4 accepts as input (a) the experimental mass spectrum, (b) the mathematical model that defines the distribution of comonomers along the chain, and (c) the process by which the oligomers subjected to MS were obtained (i.e., if they are preformed or obtained in a nonselective or selective cleavage process (see below)). The program generates the theoretical mass spectrum (see Figure l), and, if a best fit is requested, the computer code varies the parameters associated with the selected mathematical model until it finds the best match between the observed and calculated data. For this purpose the minimization subroutine MINPACKI-LMDIFI (belonging to the Argonne Library) is used.1° MACO 4 yields as output the parameters that give the best agreement and the corresponding theoretical spectrum. The difference between the observed and calculated values is expressed by means of the Hamilton agreement factorlOJ1(AF):

where IiexPtland Iicdcd are the normalized experimental and calculated abundances of the oligomers. An average AF value is usually given by MACO 4 (see tables below). However, the program also calculates separate AF values for each group of oligomers,which can be usedlOJ1to check the accuracy of the experimental data and the selfconsistency of the calculations. Furthermore, MACO 4 provides an estimate of the composition of the copolymer, independent of that deriving from the statistical modeling calculations. 3. Effect of the Partial Degradation Process on the Mass Spectra of Copolymers When a copolymer sample is subjected to partial degradation, a mixture of oligomers is formed and the intensity of each peak in the mass spectrum of a twocomponent (AB) copolymer is the sum of contributions due to oligomers having the same overall formula but different comonomer sequences. For instance, the intensity of MS peaks associated with the tetramers AzBz and A3B is

where I X X X Xis the probability of finding the comonomer sequence XXXX (where X may be A or B) in the mixture of oligomers. The experimental intensities must therefore be pointwise deconvoluted to obtain the quantities Ixxxx, which are directly proportional to Rgxxx, the probability of finding the sequence of comonomers XXXX in the nondegraded copolymer chain:

where a are proportionality factors which depend on the cleavage mechanism. Equation 3 implies that the mass spectral peak intensity depends on the distribution of comonomers along the copolymer chain. The initial assumption on which we developed1@12algorithms to match the experimental mass spectra to those theoretically calculated was that the partial degradation of the copol-

4266 Montaudo and Montaudo

Macromolecules, Vol. 25,No. 17, 1992 Ph

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Figure 1. (a)Positive-ion FAB mass spectrum of the oligomers generated in the photolysis of copolyamide III.17 (b)Theoretical mass spectrum generated by statistical modeling. ymer chain occurs by a rigorously nonselective cleavage process. If all B values in eq 3 are equal to 1, the relationship Ixxxx = G x x x (4) holds for each sequence XXXX. Nonselective cleavage may occur in the pyrolysisof olefin and vinyl copolymers2J6 and has been actually observed when a condensation copolymer is subjected to methanolysis,11J2hydrolysis,l9 and aminolysis.l8 Instead, a selective cleavage may occur in condensation copolymers bearing two different functional groups in the chain (for instance, amide and ester groups). Furthermore, thermal,21p h ~ t o l y t i cand , ~ ~ozone20 cleavage may produce totally selective degradation processes along the copolymer chain. In fact, as illustrated in Figure 2, the cleavage process splits in two parts one of the comonomers, leaving intact the other. In the latter case, the probability ( R L Y B B ) of finding the sequence of comonomers AABABBB in an infinite copolymer chain and the probability (IAABABBB) of finding the sequence AABABBB in a heptamer are not the same (even the total number of MS peaks in the mixtures of oligomers is different from the case of nonselective cleavage, as shown in Table I, column 3), and a relationship between the two quantities has to be found for each specific case. In the following sections we shall derive equations for the probability of finding an oligomer sequence in the infinite copolymer chain and then we shall see how to modify them to match the conditions defined by the occurrence of a selective cleavage process. 4. Sequence Distributions in an Infinite Copolymer Chain In our analysis we consider three different distributions of monomers along the copolymer chain, namely, firstorder Markovian, Bernoullian, and sequential. We computeforeachdistribution Rixxx, theprobability offinding the sequence of comonomers XXXX in the infinite chain. A Markov distribution is completely defined when the transition probability matrix (Pmatrix) is given. The P

a ) Pyrolysis

- CH,-CH,-

C Q 4 $ & O ~ C O ~ C Q - Q

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p-Hydrogen

Tronrfrr

I

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A *COS

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---*-t

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--+

cw>-

cn i;

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Ph

CH,-

CH I ** Ph

Ph

coon

+

onc

- cn, C CH,-

cn+--Ph

Figure 2. Examples of totally-selective chain cleavageprocesses.

matrix is a square matrix of dimension N, where N is the number of different components that occur in the chain. Each matrix element Pij describes the probability of fiiding the component i after the component j . For example, a Markov distribution for a two-component copolymer (AB) is defined when the four P matrix elements PAA,P m , PBA, and PBBare given. The P matrix elements Pi, cannot assume arbitrary values. In fact, they must be positive

Macromolecules, Vol. 25, No. 17, 1992


and they must satisfy the following property:

10 as

PU

+ PAB = 1

PBA + PBB

1 (5) The probability of finding the hypothetical monomer sequences AABBA, AABAB, and BABBA is given by

(11) = (1- PBB1-I If the chain follows a Bernoullian distribution, substituting eq 8 into eq 10 one obtains

The quantities RZ and R i (the probability of finding the sequence A or B in the infinite chain) in eq 6 are given by

( n ~=) (1- &)I-' ( n ~=)(1- p(B))-' (12) If the chain follows a sequential distribution, numberaverage lengths can be evaluated using the definition in eq 10. Sequence Distributions in a Finite Chain (Oligomers). In the case of low molecular weight copolymers and in cases where the copolymer chain cleavages occur by a nonselective process, the u factors in eq 3 are equal to 1and the identity in eq 4 holds for each sequence XXXX. This means that, in the cases specified above, eqs 6-9 can be directly applied to predict the MS peak intensities for copolymers that follow Bernoullian and Markovian distributions. We report here the pertinent equations in the final form actually used in our computations (MACO 4). In the case of Bernoullian distributions,lO from eqs 8, 2, and 4 it follows

R i = PBA/(PAB+ PBA) R i = P A B / ( P+~PBA) (7) RZ was called SA in the previous paper,lo since Ri also represents the probability that the chain starts with A. We now compute R G , , for a Bernoullian distribution. A Bernoullian distribution is completely defined when the molar fraction of A, p(A), and the molar fraction of B, p(B),are given. The probability of finding the hypothetical monomer sequences AABBA, AABAB, BABBA, A, and B is given by

We now compute RGxxx for a sequential distribution. This distribution is defined when the repeating sequence (Goo) is given. In a first approximation, RGxXx can be computed generating an artificial chain by repeating a large number of times the given sequence Goo and then using a "search-substring-in-string"procedure23to compute the number of times, NXXXX, that a given sequence XXXX occurs in the chain. We shall refer to this method for deriving R" values for a sequential distribution as the "rate-of-occurrence"method. Accordingly, the probability of finding the sequence XXXX in the artificial chain is given by

Rixxx = Nxxxx/L (9) (here L is the length of the artificial chain expressed in number of comonomer units present). Another quantity related to the sequence distribution of an infinite chain is the number-average length of like monomers,l3 which is a measure of the degree of "blockiness" of a copolymer chain. In the two-component copolymer example the number-average lengths of component A, ( n ~ )and , of component B, ( n ~ )are , defined by

If the distribution of comonomers along the copolymer chain is Markovian, eqs 6 and 7 can be used to rewrite eq

(nA) = (1 - PU)-'

IB,

(nB)

= P(B)P(B)P(B)

(13)

In the csae of Markovian distributions,10from eqs 6,7, 2, and 4 it follows

4268


Montaudo and Montaudo

The theory can be extended to the case of sequential copolymers. In fact, from eqs 9, 2,and 4 it follows

Table I1 Experimental. and Calculatedbe Relative Amounts of Preformed ET-TX Oligomers (ET= Ethylene Terephthalate, TX = Ethylene Truxillate)*s calcdb oligomer

(ETh (ET)(TX) (TX)z (ET)z(TX) (ET)(TX)z (ET)3(TX)

(TX)3 (ET)z(TX)z (ET)(TX)3

(ET)3(TX)z (TW4 (ET)z(TX)3 (ET)(TX)d

(ET)3(TX)3 AFf

IB, = NEEEIL

(15)

Equations 13,14,and 15 constitute a complete group of equations, valid in the case of nonselective cleavage, which can be used to compute the theoretical mass spectrum of a copolymer that follows Bernoullian, Markovian, or sequential distributions. Before going further with the theory, it seems appropriate to illustrate (in the following sections) a few examples where the equations derived up to now are used to model the experimental mass spectra. 5. Microstructure Determination in Low Molecular Weight Copolymers The analysis of low molecular weight copolymers by MS is simpler than that of high molecular weight copolymers since their low weight avoids the necessity to subject the sample to partial degradation prior to the MS analysis. Furthermore, their mass spectra depend solely on the variables that describe the distribution of monomers along the chain and not on the cleavage process, rendering the simulation procedure straightforward. In many copolymerization processes, low molecular weight copolymers are formed together with high molecular weight copolymers (bimodal distributions) and both species have the same microstructure9J9 (Bernoullian, Markovian, sequential, etc.). Therefore, the analysis of (performed) low molecular weight species by MS allows the determination of copolymer microstructure. We report here two examples8plgof microstructure determination from preformed oligomers. The structure of oligomers can be easily assigned in the original mass spectra8J9and, starting from the experimental intensities, eqs 13-15 can be directly applied.

In the first example, mixtures of low molecular weight oligomers of a copolyester containing ethylene terephthalate units (ET) and ethylene truxillate units (TX) (structure I) were analyzed by means of FAB-MS.lg

COOCH2CHZOlos\CO +c-HzocHz.

Ph

I

To proceed to the microstructure determination, the observedl9 MS peak intensities of the 15 mass peaks cor-

m/z obsd' 401 10.6 531 26.6 593 3.4 661 11.2 723 8.8 853 9.6 915 3.9 983 3.7 1045 5.0 4.8 1175 1237 2.1 1305 2.6 1367 3.7 1497 2.4 1559 1.6

H1 12.1 24.2 3.1 12.1 9.5 9.5 4.4 3.1 6.6 4.4 3.3 1.1

3.3 1.6 1.6 0.117

calcdc H2

calcdd H3

calcde H4

0 48.4 0 0 12.6 12.6 0 0 16.5 0 4.1 0 4.1 0 1.6

4.8 38.7 0.5 4.8 12.1 12.1 2.6 5.0

19.4 9.6 8.0 19.4 4.5 4.5 3.5 8.0 3.7 3.5 2.8 5.7 2.8 2.6 1.6

0.874

11.1

2.6 3.9 0.1 3.9 0.4 1.6 0.503

0.652

Relative intensities of the (M-) ions in the FAB ma88 spectrum taken from Table 111,column 6 in ref 19. Computed for model H1 using eq 13. Computed for model H2 using eq 15. Computed for model H 3 using eq 14. e Computed for model H 4 using eq 14.f Agreement factor computed using formula 1. a

responding to oligomers ranging from dimers up to hexamers (shown in Table 11)were given as input to MACO 4. Since the copolyester composition is knownlgto be p(ET) = 0.50, p(TX) = 0.50, theoretical mass spectra were generated for four different chain models, H1,H2, H3, and H4, which all correspond to the above composition (Table 11). Model H1 is a pure Bernoullian distribution defined by p(ET) = 0.50, p(TX) = 0.50 with associated number-average lengths (nm) = 2 and ( ~ T X =) 2. Model H 2 is a pure alternating (ET-TX), distribution with associated number-average lengths (nm) = 1 and ( ~ T )X = 1. Model H 3 is a pure Markovian distribution having P matrix elements P ~ , E= T 0.20,P ~ , T=x0.80,PTX,ET = 0.80, and PTX,TX = 0.20 with associated number-average = 1.25 and ( ~ T x )= 1.25. Model H 4 is a lengths (nm) pure Markovian distribution having P matrix elements Pm,m = 0.80, PET,= = 0.20,P T X ,=~0.20,and PTX,TX = 0.80 with associated number-average lengths ET) = 5 and (wx) = 5. In Table I1the theoretical peak intensities are reported for the four selected models. Model H1 gives the best agreement (AF = 11.7%; Table 11), and, it is therefore concluded that the TE-TX copolyester is best described by a pure Bernoullian distribution with (nm)= 2 and ( ~ T x )= 2,as would be expected from the method used in the synthesis.lg In another case: low molecular weight oligomers, formed during ther synthesis of a copolymer containing units of methyl methacrylates (MA) and units of butyl methacrylate (BA), were analyzed by laser desorption Fourier transform MS.8 To find the distribution of MA and BA units along the chain, the observed intensitiess of the 58 mass p e d s corresponding to oligomers ranging from nonamers up to 15-mers (reported in Table 1s)were given as input to MACO 4 and three different chain models, K1,K2, and K3,were considered (Table 1s). Model K1 is a pure sequential distribution having a ten-monomer repeating unit: (MA-MA-BA-MA-MA-BA-MA-MA-MA-BA),. This particular sequence was selected because its associated composition (p(MA) = 0.70,p(BA) = 0.30)coincides with the composition given by the copolymer's manufacturer.8 Model K 2 is a pure Bernoullian distribution defined by p(MA) = 0.74,p(BA) = 0.26,which gave the

Macromolecules, Vol. 25, No.17, 1992

Statistical Modeling of Mass Spectra of Copolymers 4269 CO-Oft

-(CH-CH21

CH-CH2-

0.85

CH3

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peaks corresponding to monomers in Figure 3a were not best agreement factor in the minimization process among included in the simulation. In fact, for all the other groups pure Markovian distributions. Model K3 is a pure Marof oligomers (from dimers to heptamers), the volatility kovian distribution having P matrix elements Pm,m = ~ . ~ ~ , P ~ , B A = ~ . ~ ~ , P B A , ~ = O . ~ O , ~ ~ ~ P B A , difference B A = O . in ~ Othe ( WMS ~ ~is~ negligible. Instead, for the two monomers HB and HV the difference in volatility cannot an associated composition p(MA) = 0.69 and p(BA) = 0.31) which gave the best agreement factor in the minimization be neglected, and thus the ratio of their peak intensities process among pure Markovian distributions. (IHB/IHV = 75/25) does not reflect the copolymer comConsidering that a radical copolymerization process was position (IHB/IHV = 85/15).11 Also, peaks beyond m/z = 643 in Figure 3a were not included in the calculation used to synthesize this MA/BA copolymer! some qualibecause they are knownll to be scarcely reproducible. tative predictions can be made based on copolymerization theory:15 the pure Markovian distribution (model Besides these straightforward examples, in some instances it is necessary to consider more complex models K3) is expected to yield the best agreement factor, while such as the case of binary mixtures of copolymers and/or the agreement with experimental intensity data for the pure sequential (model K1) and for the pure Bernoullian the case of uncertainties in the structural assignment of MS peaks. Therefore, prior to discussing these cases, we (model K2) distributions is expected to be poor. These shall derive the theory necessary to analyze these systems. predictions are thoroughly confirmed by our computations (Table lS), which yielded the following values: for model If the sample to analyze is a binary mixture of two copolymers, say C1 and C2, having the same comonomers K1, AF = 1.273; for model K2, AF = 0.241; for model K3, but different sequence distributions along the two chains, AF = 0.092 (( n m ) = 4.5, ( ~ B A = ) 2). the resulting MS peak intensities shall be the weighted 6. Oligomers Generated by Nonselective Chain sum of the contributions due to C1 and C2, and, therefore, Cleavage Processes eq 2 cannot be directly applied. If we call X the molar fraction of copolymer C1 in the mixture and (1- X) the In the case of oligomersgenerated by nonselective chain molar fraction of copolymer C2, we can rewrite eq 2 as cleavages of copolymers, eqs 13-15 can be directly applied to obtain the microstructure. Some examples of analysis IA,B, = x ( I L B B + + I L B A + I L B + ILm + of mixtures of oligomers generated by methanolysis and pyrolysis of microbial copolyesters(both nonselectivechain &h)+ (1- x ) ( I ~ B B + I ; B ~ + I ~ B B A + I ~ W + processes in these specific cases) have been previously However, the best fit copolymer composiGAB* + GBA.4) tions were calculated separately for each group of oligoIA,B = X U L A B + I L B A + Ihh + I,-)+ 1 mers. As a matter of fact, previous computer programs (MACO ill and MACO ,la) do not allow display of the full (1- X ) ( I k B + F u B A + &u + G-1 (16) mass spectra calculated by statistical modeling. In Figure 3a is reported the negative ion FAB mass spectrum of the partial pyrolysis residue of P(HB-co-l5%HV) (ref 11, where I;,,,! and GxXx are the contributions to the MS sample 3), and in Figure 3b is displayed the mass spectrum peak intensities deriving from copolymer C1 and from calculated for a Bernoullian distribution (eq 13) by MACO copolymer C2. To derive some explicit expressions from 4. We may notice that the observed intensities of MS eq 16, we must specify the type of distribution followed



by chains C1 and C2. If copolymer C1 follows a Bernoullian distribution defined by molar fractions pl(A) and pl(B) and copolymer C2 follows a sequential distribution, then, inserting eqs 8, 4,and 9 into eq 16,one obtains IA

= Xpl(A) + (1 - X)(NAIL)

the given sequence XXX occurs in the chains C1 and C2. If copolymer C1 follows a Bernoullian distribution defined by molar fractions pl(A) and pl(B) and copolymer C2 follows a Bernoullian distribution defined by molar fractions p2(A) and p2(B), eq 16 takes the well-known11 form

IB,

= Xpi(B)p,(B) + (1 - X)pz(B)p,(B)

(19)

Number-average lengths ( n ~and ) ( n ~ for ) a binary mixture of copolymers are the weighted sum of numberaverage lengths for the two pure copolymers. Therefore, to evaluate ( n ~and ) ( n ~ one ) , must first compute the number-average lengths ( n i ' ) , ( n i ' ) ,( n f ' ) ,and (ng') for copolymers C1 and C2 making use of eq 12,10,or 11 and then combine these quantities according to the formulas

+ (1- x)("2') (nB) = X ( n i ' ) + ( l - X ) ( n g ' ) (nA) = x(n:')

IB,

= Xpi(B)pl(B)pi(B)+ (1 - X)(NM/L)

(17)

If copolymer C1 follows a sequential distribution with repeating sequence Go1 and also copolymer C2 is described by a sequential distribution with repeating sequence G02, then, using eq 9,eq 16 becomes LI, /"= , "x

+ (1- X ) N ~ ) I L

I B , = XN& ' !,/L

+ (1 - X)NgA/L

IA, = XN:L/ L + (1 - X ) N c /L

(20) The molar fractions of comonomers A and B in the binary mixture are computed by inserting the values for IA and I B obtained from eq 19, 17,or 18 in the expressions

The theoretical MS peak intensities obtained using eqs 13-19 are valid when all peaks in the mass spectrum belong to the same mass number series, i.e., when all peaks correspond to oligomers A,B,, which start and end with the same chemical groups. Therefore, once a theoretical MS spectrum (ICdCd) is generated using eqs 13-19, a comparison between theoretical and measured MS intensities can be readily done by computing the agreement factor (eq 1). However, often the mass spectrum displays peaks associated with two different mass series, say S1 and S2, which correspond to the same oligomer but have different end groups. Therefore, the total number of MS peaks that can be seen in the mass spectra of AB copolymers increases as 2(n + 1) (Table I, column 4). In these cases, one must first compute the theoretical MS intensities F1,cdcd and F2,caicd for the two series and then evaluate separately the agreement factors AFS1and AFS2 for the two series using eq 1. The total agreement factor is obtained as a combination of AFS' and AFS2: AF = (l/q,,,)[qlAFS1 + qzAFS21 q1

=

(~(1~1~exptl)2)1/2

i

IB,

= xMiAB/L

+ (1 - X)N$AB/L

(18)

where NgLxx and N$Lxx are the number of times that

q2 =

(C(I;1,exptl)2)1/2 1

(22) where F1,expti and F2vexpt1 are the intensities of MS peaks belonging to the series S1 and 52. In Table 2 s is reported an example of this type (discussed below). Copolymer mass spectra displaying two different mass series may show a further complication. In fact, it can happen that the MS peak seen at mass number p has an uncertain structural assignment since it can be associated with the oligomer A,B, belonging to series S1 or associated with


Macromolecules, Vol. 25, No. 17,1992

“e

CH3

O - C O + ( O ~ O - C O + CH,

a)

476

loo

50

05

05

FAB Mass Spectrum

1 -

x 10

, 730

451 396 412

222

I

666

’Io2

848

1213

333

1 5 1 0 1582 1 6 2 1

6e

AB 476

loo

50

1-

b)

I

Mass Spectrum Calculated by Statistical Modeling

B, A,B,

B4

’

866

I1

i i z o 12”A B 12;11 123(1

959 I

I

AB.

AB,

AB4

1610

l d b 9 1374 1 1

1621

I

Figure 4. (a) Positive-ion FAB mass spectrum of the oligomers generated in the partial aminolysis of RE-BP copolycarbonate IIa.l* (b) Theoretical mass spectrum generated by statistical modeling. the oligomer ArB, belonging to series S2. A deconvolution procedure must be applied in these cases, since the intensity of the peak associated with mass number p is the weighted sum of the contribution from the isobaric oligomers APBqand ArB,. Therefore, one must compute the theoretical intensities F1*calcd and F2*cBlcd for series Sl and S2 and then combine the intensities of isobaric oligomers using the formula fl+S2,calcd I tfl,c&d + (1- tlf2,calcd (23)

IIa CH,

IIb

where t is an unknown parameter. In this case, the copolymer compositionwill be a linear combination of the F19dd, and compositions associated withP1+sz~~d, and it is also necessary to modify eq 22 for the agreement factor in the following way:

IIC

where F1+S2*expt1are the intensities of MS peaks belonging to both series S1 and 52 and where AFS1+S2 is obtained in eq 1. by inserting P + S z @ x p t l and F1+S2,dcd To illustrate some complex systems, we have selected an aromatic copolycarbonate (structure IIa) containing resorcinol units (RE) and Bisphenol A units (BP). The copolymer was synthesized by interfacial polymerization18 with the purpose of obtaining an (exactly) alternating copolymer. to check its microstructure, the copolycarbonate was amminolyzed using piperidine, and the FAB mass spectrum of the mixture of products obtained was recorded. Experimental MS datal8 are reported in Figure 4a and in Table 2S,and, from inspection of these data, it appears that several peaks do correspond to sequences containing blocks of two, three, and four consecutive BP units, thus demonstrating that the copolymer is not exactly alternating.18 Peaks in the MS spectrum can be grouped18 in two mass series: series S1 (structure IIc) has one piperidine end group and series S2 (structure IIb) has two piperidine end groups.

To proceed to the microstructure determination, the observed18 MS peak intensities of the 22 mass peaks corresponding to oligomers ranging from dimers up to heptamers (Table 2S), were given as input to MACO 4, and eq 22 was used to compute the agreement factor. Since the amminolysis cleavage is a nonselective process, eqs 13-19 were used to generate theoretical intensities and five different chain models, M1, M2, M3,M4, and M5, were considered. Model M1 is a pure alternating (RE-BP), distribution, which implies a composition p(RE) = 0.50, p(BP) = 0.50. Model M2 is a pure sequential (RE-BPRE-BP-BP), distribution, with an associated composition p(RE) = 0.40, p(BP) = 0.60. Model M3 is a Markovian distribution having P matrix elements” P = 0.00, Pmap = l.oO,pBp,m= 0.76, andPBp,Bp = 0.24 (with an associated composition p(RE) = 0.43,p(BP) = 0.57), which gave the best agreement factor among pure Markoviandistributions in the minimization process. Model M4 is a mixture of two pure sequential copolymers containing 86% alternating (RE-BP), and 14% sequential (RE-(BP)m), distribution with an associated composition p(RE) = 0.44, p(BP) = 0.56. This model takes into account spurious sequencescontaining consecutive BP units by introducing a second chain containing long (BP),blocks. The value X = 0.86 gave the best agreement factor in the minimization

4272


Montaudo and Montaudo

Table I11 Experimentals and Calculatedhe Relative Amounts of the Partial Pyrolysis Products from an AN-ST Copolymer (AN = Acrylonitrile, S T = Styrene)lG oligomer

miz

obsdo 13.9 0 0 11.1

4.6 0 0

12.0 0.4 0 1.8 8.8 0.3 0 6.9 0.9 0 9.2 0.3 0 0

0.5 4.2 0

calcdb G1 7.8 1.7 0.2 7.8 4.3 0.9 3.9 5.8 2.6 0.8 4.3 4.3 1.2 1.7 4.3 3.2 2.6 2.9 1.1 0.4 0.9 2.3 1.7 1.1

calcd' G2 12.5

calcdd G3 12.4

calcd' G4 13.3

0 0

0 0 12.4 1.4 0 0 15.7 0 0 1.4 7.7 0 0 7.7 1.0 0.1 8.5 0.1 0 0 1.0 3.4 0

0 0 11.1 2.8 0 0 14.2 0.5 0 1.5 8.4 0 0 6.7 2.0 0.1 7.6 0.3 0 0 1.0 3.7 0.1

12.5 0.1 0 0 18.5 0 0 0.1 7.9 0 0 7.9 0.1 0 10.6 0 0 0 0.1

3.5 0

oligomer

m/z

obsda 0 2.8 1.8 0 0 0 0.1 6.9 0 0.4 3.7 0 0 0 4.6 0.9 0 4.2 0 0 0.5

calcdb G1 1.2 1.7 1.9 0.7

calcdc G2

calcdd G3

calcde G4

1.3 1.2 0.1 0.7 1.1

0 3.5 0.1 0 0 0 0 9.1 0 0.1 4.2 0 0 0 4.2 0.1 0 5.5 0 0 0.1

0 3.4 1.1 0 0 0 0 7.0 0.1 1.0 4.0 0 0 0 4.0 0.7 0.1 4.1 0 0 0.7

0 2.8 2.0 0 0 0 0.1 6.1 0.5 1.0 4.4 0.1 0 0 3.2 1.3 0.1 3.5 0 0 0.6

0.586

0.323

0.185

0.111

0.1

0.3 1.1

2.2 1.3 1.9 1.9 0.7 0.1 1.1 1.9 1.1

a Relative intensities of the (M+) ions from the field ionization mass spectrum taken from Figure 3c in ref 16. Computed for model G1 using eq 13. Computed for model G2 using eq 15. Computed for model G3 using eq 17. e Computed for model G4 using eq 14. f Agreement factor computed using formula 1

process. Model M5 is a binary mixture of copolymers containing a 77% sequential (RE-BP), and a 23% Bernoullian distribution with composition p2(RE) = 0.27, pz(BP) = 0.73. The overall composition of this mixture is p(RE) = 0.45, p(BP) = 0.55. The values X = 0.77 and p2(RE) = 0.27 gave the best agreement factor in the minimization process. The results of these calculations (Table 2s) specify that the distribution of monomers along the copolycarbonate chain is best described by a pure Markovian distribution (model M3; AF = 5.9%),which accounts for the presence of consecutive sequences of BP units found in the mass spectrum18 by setting PBP-BP (the probability of finding unit BP after unit BP) to a value different from zero. The number-average length of BP sequences is (ngp) = 1.3, and the average number of RE-BP bonds over 100 repeating units24in the chain is given by 1ooR&.-Bp = 43. In Figure 4a is reported the FAB mass spectrumla of the partial aminolysis residue of copolymer IIa, and in Figure 4b is displayed the maas spectrum calculated by statistical modeling (Table 2s). As discussed above for P(HB-co15%HV)(Figure 3), also in the present case the observed intensities of MS peaks corresponding to monomers in Figure 4a were not included in the simulation. In another example, we analyzed a copolymer obtained by free-radical copolymerization, an AN-ST copolymer (AN = acrylonitrile, ST = styrene), studied by means of pyrolysis-field-ionization MS.16 To proceed to the microstructure determination, the observed16 MS peak intensities of the 45 mass peaks corresponding to oligomers ranging from pentamers up to 12-mers (shown in Table 111) were given as input to program MACO 4. The pyrolysis cleavage was considered to be a nonselective process, and four different chain models, G1, G2, G3, and G4, were considered. Model G1 is a pure Bernoullian distribution defined by p(AN) = 0.50,p(ST) = 0.50, which gave the best agreement factor among Bernoullian distributions in the minimization process. Model G2 is a pure alternating (AN-ST),, distribution. This model was

chosen because it predicts the systematic absence of MS signals at mlz = 422,469, 520, 573, 632, 677,991, which is a relevant feature of this mass spectrum16 (Table 111). Model G3 is a binary mixture of copolymers containing 96 % of an alternating (RE-BP), and 4% of a Bernoullian distribution with composition pz(AN) = 0.50, pz(ST) = 0.50. The value X = 0.96 gave the best agreement factor in the minimization process. Model G4 is the pure Markovian distribution having P matrix elements PAN,AN = 0.11, PAN,ST = 0.89, PST,ST = 0.11, and &,AN = 0.89 (with an associated composition of p(AN) = 0.50,p(ST) = 0.50), which gave the best agreement factor among pure Markovian distributions in the minimization process. Table I11 reports the theoretical mass spectra for the four models. The results show that the chain is best described by a Markovian distribution (model G4; AF = 11.1%). These results are in agreement with free-radical copolymerization kinetic theory,15which predicts that the microstructure of those copolymers is seldom described by a pure Bernoullian or sequential disrtibution but usually is first order Markovian (in fact, agreement factors for models G1, G2, and G3 are poor). The Markovian distribution predicted by the theory15 has P matrix elements given by PAA=

rAfA fB + rAfA

rBfB

PBB

= f A + rjB

(25)

where r A and rg are the copolymerization reactivity ratios for radical A* and radical B* and f~ and fB are the molar fractions of the two comonomers in the feed. For the AN-ST 5050 copolymer, the quantities ?-AN, r s T , fAN, and fST are available from the l i t e r a t ~ r e . ~ ~ Inserting these experimental values in eq 25, we obtain PAN,AN = 0.114 and PST,ST = 0.114, which are in excellent = 0.11 agreement with the values P A N , A N= 0.11 and PST,ST derived from the analysis of the mass spectrum (Table 111). As mentioned above, uncertainties in the structural assignment of MS peaks increase the difficulty of mass



Table IV Experimental. and Calculated&dRelative Amounts of the Partial Pyrolysis Products from an AN-ST Copolymer (AN = Acrylonitrile, ST = Styrene)l6 deprotonated oligomer

protonated oligomer

mlz

obsd"

calcdb

calcdC

4.6 4.6 3.5 3.8 0.2 2.7 2.9 1.0 4.4 4.0 0.5 0.04 0.5 2.3 2.7 0.2 2.2 2.2 0.9 2.8 3.1 0.5 0.9 1.7 1.6 0.2 1.5 1.5 0.6 0.1 1.7 1.9 0.3

0.6 2.9 5.8 1.2 0.1 5.9 3.1 0.7 4.1 2.1 0.3 0.01 3.1 3.4 1.2 0.1 3.4 2.3 0.6 2.9 1.4 0.3 2.3 2.0 0.9 0.3 2.0 1.6 0.8 1.4 1.9 1.7 0.7

4.5 4.8 3.4 3.8 0.1 2.8 3.0 0.9 4.3 4.0 0.4 0.1 0.4 2.2 2.8 0.2 2.4 2.2 0.8 2.9 2.9 0.5 0.8 1.8 2.0 0.1 1.4 1.5 0.5 0.4 1.6 1.9 0.3

deprotonated oligomer

protonated oligomer

mlz

obsda

calcdb

calcdc

835 837 839 841 886 888 890 892 894 939 941 943 945 990 992 994 996 998 1043 1045 1047 1049 1051 1096 1098 1100 1102 1149 1151 1153 1155

1.8 3.9 3.9 0.3 0.01 3.3 3.9 1.4 0.1 0.3 3.2 3.4 0.7 0.01 0.9 1.8 1.5 0.2 0.01 1.29 1.82 0.53 0.06 0.15 1.14 1.34 0.03 0.30 0.58 0.43 0.12

1.6 2.4 1.6 0.3 0.9 2.4 2.6 0.8 0.3 1.7 3.0 1.5 0.1 0.9 2.6 2.0 1.1 0.7 1.62 2.05 1.33 0.07 0.12 0.15 0.91 0.75 0.17 0.50 0.65 0.21 0.11

1.1 4.0 3.7 1.0 0.1 3.4 4.1 1.3 0.1 0.5 3.3 3.3 0.6 0.8 0.7 1.8 1.5 0.3 0.01 1.16 2.01 0.51 0.01 0.05 1.10 1.62 0.02 0.37 0.57 0.38 0.10

0.439

0.037

*

" Relative intensities of the ions from the electron impact mass spectrum taken from Figure 4 in ref 16. Computed using eqs 14 and 26 with = 1.0, PAN.AN = 0.12, PAN,ST = 0.88, PST,AN = 0.92, and PST,ST = 0.08. Computed using eqs 14 and 26 with X = 0.5, PAN,AN = 0.12, = 0.92, and PST,ST = 0.08. Agreement factor computed using eq 24. PAN,ST0.88, PST,AN

x

spectral interpretation because the contributions of all isobaric structures that determine the total peak intensity must be considered. Such equivocal structural assignment of MS peaks occurred when the above-discussed AN-ST copolymer was analyzed by means of direct pyrolysiselectron-impact (EI-MS).16 The resulting ELMS6shows signals up to M-mers, which can be grouped in two series which correspond, respectively, to deprotonated and protonated AN-ST oligomers (Table IV). Most mass peaks can be assigned to both series (as shown in Table IV), and, therefore, these MS peaks have an uncertain structural assignment. Since the intensities of these MS peaks derive from two contributions, the calculation of theoretical MS intensities can be performed by computing the contribution due to deprotonated oligomers, zdeprot, and the contribution due to protonated oligomers, Zprot, and by combining these values using the formula

= XIdeprot + (l- x)zprot (26) Intensities Ideprot and Iprotin eq 26 depend on the distribution of comonomers along the AN-ST copolymer chain. The quantities X and (1- X) in eq 26 describe the probability of forming a deprotonated or protonated oligomer. These probabilities should be equal (X = 0.5), since both species derive (in the proposedl6 ion fragmentation reaction scheme) from the same homolytic scission.l6 To use eq 26 to simulate the AN-ST copolymer mass spectrum, the chain was assumed to follow Markovian statistics (accordingto the results of the previous analysis), and the thermal cleavage process was assumed to be nonselective. The observed16 MS peak intensities of the 64 mass peaks corresponding to oligomers ranging from tri'tot

mers up to 15-mers (shown in Table IV) were given as input to MACO 4, and a minimization was performed to PAN,ST, PST,AN, and search for the values of X, PAN,AN, PST,ST which identify a minimum in the agreement factor (eq 24 was used to compute the AF). These values were found to be X = 0.50, PAN,AN = 0.12,PAN,ST = 0.88, PST,AN = 0.92,and PST,ST = 0.08 with an associated AF = 3.7% (Table IV). Comparing the P matrix elements obtained from the simulation of the E1 mass spectrum in Table IV with those found for the same AN-ST copolymer analyzed by FI-MS (Table 111),it can be noticed that the two sets of values are very close and that, furthermore, they are in excellent agreement with the values obtained from kinetic~ through ~ ~ eq 25 (see above). Also, the value X = 0.50 found for this system (Table IV) indicates that the probabilities of forming a protonated and a deprotonated oligomer are equal, as required by the ion fragmentation process.16 If a value X = 1.0 is selected for the simulation, a much worse agreement is obtained (Table IV, AF = 43.9% 1. Uncertainties in the structural assignment of MS peaks can also occur when, in a copolymer, molecular weights of the two comonomer units differ by one mass unit and both protonated and nonprotonated species are present. This is the case of a copolymer containing acrylonitrile (AN) and butadiene (BU) units studied16 by direct pyrolysis-field ionization MS. In this system it happens that the BU units (mlz = 54) appear always as nonprotonated species, whereas the AN units (mlz = 53) may appear as protonated and nonprotonated species. Therefore, protonated AN species are isobaric with BU species (Table 3s).This copolymer produced a mass spectrum containing


Macromolecules, Vol. 25,No.17, 1992

peaks ranging from monomers up to 16-mers.16 The inspection of the mass spectrum%eveals that the intensity of MS peaks corresponding to AN protonated oligomers (observed at m/z 107, 160, 213, 266, 319, and 372) are incompatible with the overall 2:l molar ratio of comonomers in the copolymer,16and, therefore, it was inferred16 that some AN homopolymer ought to be present, mixed with the copolymer. This means that the observed MS intensities derive from three contributions-the contribution of AN-BU nonprotonated oligomers (referred to &9 IAN-BU the contribution of AN-BU protonated oligomers (refered to as and the contribution of AN protonated homopolymer sequences (referred t o as Zi:r)-and that the resulting mass spectrum, Iht, is the weighted sum of the above three intensities: - T IAN-BU AN-BU + T Ihomo 'tot

-

1 nonprot +

Tdprot

(27)

3 prot

In eq 27 T1 + T2 is the molar fraction of AN-BU oligomers, T3 is the molar fraction of AN homopolymer, and andIELBUdepend on the AN-BU chain intensities 1::;: model. For our numerical simulation we considered 101 peaks in the mass spectrum (Table 35) corresponding to oligomers ranging from dimers up to 12-mers. The chain was assumed to follow the Markovian chain model, the cleavage processwas assumed to be nonselective,and MACO 4 was used to find the values of T I , T2, T3, PAN,BU, and PBU,AN that gave the best agreement. These values are T I 0.55, Tz = 0.37, T3 = 0.08, PAN,BU = 0.80, and PBU,AN = 0.39 and imply that the sample analyzed16 is a binary mixture of a 92% Markovian AN-BU copolymer and an 8% AN homopolymer with a resulting overall Bu/AN molar ratio of 62:38. The agreement factor (AF = 8.9%1, evaluated using eq 24, is reported in Table 35 together with the resulting mass spectrum. For comparison, other theoretical spectra, obtained using eq 27 with T I = 1.0, T2 = 0, and T3 = 0 and with T I = 0.92, T2 = 0, and T3 = 0.08, are also reported, but the AF obtained with these values is much poorer (Table 3s).

7. Composition of Copolymers by Direct Analysis of Their Mass Spectra The peak intensities displayed by the mass spectra of copolymers contain direct information about the relative abundance of comonomers, and a copolymer composition estimate can be directly extracted from the analysis of the mass spectra26 without making any hypothesis on the distribution of comonomer units present in the copolymer. In the case of a four-component copolymer (ABCD), it is possibleto make an estimate of the copolymer composition c ~ c ~ from c ~ acknowledge ~ of the x-mer peak intensities, IA,B.c,D, using the formulas

X

x6x,q+nIB,D,] q=o

1

*

x

x

X

,

x

x

X

where flX is the sum of x-mer intensities and 6 is the Kronecker symbol (defined by bii = 1 if i = j and 6ij = 0 if i # j ) . If we specialize eqs 28-31 for a two-component copolymer and for x = 1, x = 2, x = 3, and x = 4, we obtain

Equation 32 implies that each oligomer group (i.e., dimers, trimers, tetramers, etc.) provides an independent estimate of the copolymer composition. Formulas in eq 32 have been used to compute the composition of the copolymer sampleslisted in Table V, starting from the peak intensities published in the original mass spectra of all the copolymers analyzed here.8J1J2J6-21 From inspection of Table V two different behaviors can be noticed. In the cases where the reported mass spectrum refers to preformed cooligomers8J9 (Table V, columns 11 and 15) and also in the cases of copolymer samples which have been subjected to partial degradation by means of a nonselective chain cleavage processl1J6J8 (Table V, columns 2-5, 8-10, 141, the estimated compositions are roughly the same (Figure 5c,d). In these cases, one can safely compute an average copolymer composition by means of eq 33, which provides a weighted average of the data appearing in Table V for each oligomer group: (33) However, there are a few cases in Table V (columns 6,7, 12,13, and 16) where the estimate provided by each oligomer group varies continuously (Figure 5a,b). This behavior distinguishes cases where the copolymer samples have been subjected to partial degradation by a selective chain cleavage process (Figure 2). Therefore eq 32 provides only a diagnostic test of whether the copolymer sample has been subjected to a selective cleavage process but does not allow a determination of copolymer composition. Instead, in this case, copolymer composition must be computed indirectly, by finding copolymer microstructure and calculatingthe associated composition (seebelow). As a matter of fact, a selective chain cleavage favors the production of oligomers with a certain type of end group. The resulting mass spectrum is no more a transparent image of copolymer microstructure, and the correspon-

Macromolecules, Vol. 25,No. 17, 1992


dence with the sequences present in the nondegraded copolymer chain is altered. This effect is more pronounced in short oligomers so that the composition estimates associated with low c t differ sensibly from the actual composition value. 8. Oligomer Distributions Generated by Totally-Selective Chain Cleavage Processes The case of selective chain cleavage occurs when one or more than one u factor in eq 3 takes a value different from 1and, therefore, for the sequence, say XXXX, associated with that u one has

Ixxxx # R k x x (34) As a consequence, the solutions of eq 3 given by eqs 13, 14, and 15 are not applicable and a new theory has to be developed. An exact solution of eq 3 valid for all types of cleavages cannot be found (see section 9). However, an exact solution can be derived when copolymer chain cleavage is totally selective (e.g., when the mixture of oligomers is formed only by oligomers of the type AXXXXA, which start and end with the same comonomer). Assuming that only A units are cleaved (Figure 21, u factors in eq 3 can take only twovalues, namely, 0 and 1,and, therefore, we may rewrite eq 2 as

IB, = 0

(37)

In the case of a sequential chain, from eqs 9,315, and 34 it follows

IA,B,= IABBA

In the case of Bernoullian distributions, from eqs 8,35, and 34 it follows

IB, = 0

IB,

=0

Equations 36-38 constitute a complete group of equations which can be used to compute the theoretical mass spectrum of a copolymer that follows a Bernoullian, Markovian, or sequential distribution in the case of totally selective cleavage processes. It is immediately apparent that the predicted intensities of MS peaks obtained using eqs 36-38 are different from those valid for nonselective cleavages (eqs 13-15). In fact, since only A units are cleaved, eqs 36-38 predict the appearance of oligomers containing at least two A units. The above equations also predict the systematic absence of peaks associated with oligomers Bnand ABn (n= 1,2,3, ...), and this implies that the maximum number of peaks expected in the mass spectra in the case of totally-selective cleavage processes is (n- 1)(Table I). Furthermore, for an exactly alternating chain (AB),, eq 38 predicts that peaks associated with even oligomers (dimers, tetramers, hexamers, octamers, decamers, etc.) cannot appear in the mass spectrum.28 Another notable difference between selective and nonselective cleavages concerns the composition estimates. For nonselective cleavages, composition estimates c t for the oligomer groups are predicted to be equal, as can be demonstrated by substituting the intensities of the MS peaks obtained using one of the equations (131,(14), and (15) into eq 32. The result is c:’ = c; = c; = c: = c;

In the case of Markovian distributions, from eqs 6, 7, 35, and 34 it follows

(38)

c; = c$ = c:

(39)

Experimental evidence of these predictions is found in Table V and in Figure 5c,d. An analogous calculation for the totally-selective cleavage can be performed by substituting the intensities of the MS peaks generated by eqs 36, 37, and 38 into eq 32. The result is


Macromolecules, Vol. 25, No. 17, 1992 Table V Composition EstimatesP for Various Copolymers

x-mers

CHBb

CHBc

CHBd

CHBe

,OB/

cOBg

CBUh

2 3 4 5 6 7 8 9 10 11 12 13

91.5 92.0 93.0 92.9 92.6

83.0 86.0 84.8 84.2 85.5

76.5 79.3 77.8 80.0 79.5

71.5 74.0 72.3 73.6 72.8

33.3 29.3 24.4 33.3

35.1 40.0 41.2

62.4 63.6 61.1 63.2 62.5 62.5 60.5 62.6 64.1 62.7 60.5 66.7

92.3

84.6

78.4

72.9

14

@Ti

58.8 56.4 55.9 53.8 52.3 51.5 49.8 49.6 50.8 49.0 49.7

15 16 aV

62.7

52.5

@Tj

CBAk

52.2 51.6 49.9 50.4 49.7 49.6 50.3 50.4 50.0 49.5 50.5 50.1

27.7 27.3 27.8 26.1 27.0 28.1 26.0

50.7

27.5

CTPl

CTPm

$En

CTXo

100 93.3 90.6 86.2 81.4

100 83.2 75.5 70.4 66.6

43.1 44.0 42.6 42.8 43.5 42.0

50.5 51.0 49.2 49.8 49.7

43.1

50.1

cBup

100 83.3 79.3 74.2 76.0 73.5 68.5 70.4 76.4 78.6

*

Computed using eq 32. Molar fraction for comonomer HB estimated using spectral intensities taken from Table V (sample 2) in ref 11. Molar fraction for comonomer HB estimated using spectral intensities taken from Table V (sample 3) in ref 11. d Molar fraction for comonomer HB estimated using spectral intensities taken from Table V (sample 4) in ref 11. e Molar fraction for comonomer HB estimated using spectral intensities taken from Table V (sample 5) in ref 11. f Molar fraction for comonomer OB estimated using spectral intensities taken from Figure 3 in ref 21.8 Molar fraction for comonomer OB estimated using spectral intensities taken from Figure 4 in ref 21. h Molar fraction for comonomer BU estimated using spectral intensities taken from Figure 2 in ref 16. Molar fraction for comonomer S T estimated using spectral intensities taken from Figure 3c in ref 16. j Molar fraction for comonomer ST estimated using spectral intensities taken from Figure 4 in ref 16. Molar fraction for comonomer BA estimated using spectral intensities taken from Figure 2c in ref 8. I Molar fraction for comonomer TP estimated using spectral intensities taken from Table 11, column 2, in ref 17. Molar fraction for comonomer TP estimated using spectral intensities taken from Table 11,column 3, in ref 17. Molar fraction for comonomer RE estimated using spectral intensities taken from Table I in ref 18. molar fraction for comonomer TX estimated using spectral intensities taken from Table 111, column 6, in ref 19. p Molar fraction for comonomer BU estimated using spectral intensities taken from Figure 6a in ref 20. 4 Average composition computed using eq 33. (I

Ph

CALCULATED COMPOSITION - __. ~

100

t

5 K '

I

Ph

-~

OL 1

1 ._._ -

3

5

7

9

11

~

13

15

~ - ~ 17

19

OLIGOMERS GROUP

Figure 5. Composition estimates (ct) versus oligomer group (x) for various copolymer samples computed using eq 32: (a) molar (b) molar fraction of TP unih;l7 (c) molar fraction of BU fraction of BU units;lB (d) molar fraction of BA units.s c; = 1.00

(40)

We used eqs 32 and 36 to compute composition estimates c t for a Bernoullian copolymer and we plotted c t versus n for different values of the composition p(A). The resulting diagram (Figure 1s)shows that the difference c t - p(A) is never negligibly small (ci is the asymptotic value). Experimental evidence of these predictions is found in Table V and in Figure 5a,b. We proceed now to illustrate some examples of totallyselective processes. A random copolyamide containing adipoylpiperazineunits (AP)and truxilloylpiperazineunits (TP) (structure 111)was subjected to a partial photolysis process that resulted in the cleavage of the photolabile T P units contained in the copolymer main chain.'7 Such cleavagesare totally selective; in fact, oligomers produced in the degradation process were identified by FAB-MS17 and were all found to contain cinnamoyl end groups (Figure

2b). Experimental data are available17for two copolyamide samples having different AP/TP ratios: 20/80for sample 1 (Table 4s) and 50/50 for sample 2 (Table VI). To proceed to the microstructure determination, the observed17MS peak intensities (Table 4 s and Table VI) corresponding to oligomers ranging from dimers up to heptamers were given as input to MACO 4. Both samples turned out to follow a pure Bernoullian distribution: Sample 1 has a best-fit composition p(AP) = 0.20, p(TP) = 0.80 (AF = 7.3%,Table 4s) and sample 2 has a best-fit composition p(AP) = 0.50, p(TP) = 0.50 (AF = 6.2%, Table VI). Figure l a reports the FAB mass spectrum17 of the partial photolysis residue of sample 1, and Figure l b displays the mass spectrum calculated by statistical modeling (Table 4s). We may notice that monomer peaks do not appear in the experimental spectrum, in agreement with the selective character of the cleavage process. Also the reduced number of MS peaks appearing in Figure 1is due to the selective cleavage of the copolyamide chain. If the experimental MS peak intensities of sample 1are inserted in eq 32, we can obtain the estimated molar fraction of comonomer TP for the various groups of oligomers (cTp) and may see how well these values fit the theoretical predictions (eq 40). The results, reported in Figure 6, confirm the predictions and show that cTp decreases steadily when the oligomer size increases. Another totally-selective cleavage process is the ozonolysis of a butadiene/styrene (BU/ST) copolymer, synthesized by a free-radical process.20 The ozonolysis split the butadiene units into two parts (Figure 2c), and the partial ozonolysis residue was analyzed by FAB-MSZ0To proceed to the microstructure determination, the observed20 MS peak intensities of the 41 mass peaks cor-



Table VI

IN TENWTY

Experimental. and Calculated&*Relative Amounts of the Partial Photolysis Products of an AP-TP Copolyamide (AP = Adipoylpiperazine, TP = Truxilloylpiperazine)~7 oligomer mlz obsdaFAB-MS calZ tale calcd 21.3 21.3 21.3 21.3 (TPh 347 0 0 0 0 (AP)z(TP) 393 0 0 0 0 (AP)4 439 24.0 20.0 16.0 20.1 (AP)(TP)z 543 0 0 0 0 (AP)3(TP) 589 0 0 0 635 0 W ) 5 20.0 24.0 19.8 16.0 (TP)3 693 6.5 9.9 6.9 2.0 (AP)z(TP)z 739 0 0 0 0 (AP)4(TP) 785 0 0 0 0 64% 831 13.2 13.8 13.2 14.1 (AP)(TP)3 889 2.1 1.2 0.6 0 (AP)~(TP)z 935 0 0 0 0 (AP)5(TP) 981 1039 7.0 4.4 6.9 9.9 (AP)z(TP)3 1085 4.6 4.1 3.6 2.7 (AP)4(TP)z 1131 0 0.2 0.1 0.1 (AP)(TP)4 1235 4.9 2.7 3.6 4.1 (AP)3(TP)3 1281 0 0.6 0.5 0.4 0.3 1.0 1385 0 0.3 1.0 1.0 (AP)z(TP)r 1431 1.5 0.6 AF'

0.193

0.062

0.190

"Relative intensities of the (MH)+ ions in the FAB mass spectrum taken from Table I1 column 3 in ref 17. *Computed using formula 36 with p(AP) = 0.4 and p(TP) = 0.6. 'Computed using formula 36 with p ( A P ) = 0.5 and p(TP) = 0.5. dComputed using formula 36 with p(AP) = 0.6 and p(TP) = 0.4. eAgreement factor computed using formula 1. COMPOSITION ESTIMATE 100

~

*

Q

90 r

SO 70 -

60

-_ 0

2

4

6

8

10

12

14

(ARBITRARY UNITS)

2-

I

A ? :

L

0 0

20

40

60

SO

100

MOLAR FRACTION OF ET COMPONENT

Figure 7. Comparison between experimental peak intensities (X) and the theoretical probability curve of the pentamer (ET)d(OB),generated by applying eq 36 to four samples of copolyester IV of different compositions. composition p(BU) = 0.52,p(ST) = 0.48and ( ~ B I J )= 1.28, ST) = 1.11. This model is an almost alternating distribution (number-average lengths almost equal to unity), having the same monomer ratio as model 22. The theoretical mass spectra (Table 55) reported for these three models show that the copolymer chain is best described by a Markovian distribution (model 22,AF = 6.2?6), as could be expected from free-radical copolymerization theory.I5 The molar ratio of comonomers associated with model 22 (BU/ST = 53/47) is in good agreementwith the 55/45molar ratio found by lH NMR.20 Analogous to the previous example (see data in Figure 6), we can compare the composition estimates predicted by eq 40 with the estimates obtained inserting the experimental MS peak intensities from Table 55 in eq 32. The results reported in Figure 6 do not fit as well as in the previous example, and only the general trend predicted by eq 40 is confirmed in this case. The last example considered concerns four copolymer samples (structure IV) containing different amounts of ethylene terephthalate units (ET) andp-oxybenzoateunits (P0).z1 The copolymers were examined by direct-pyrol-

18

OLIGOMERS GROUP

Figure 6. (0) Molar fraction of TP units in copolyamide 111, estimated using MS peak intensities'' reproduced in Table VI, column 3. The overlying curve (continuousline) represents the theoretical composition estimates cTp, computed using eq 32. ( 0 )Molar fraction of BU units in B6/ST copolymer, estimated using MS peak intensities" reproduced in Table 5.9, column 3. The overlying curve (continuousline) represents the theoretical composition estimates c,': computed using eqs 32 and 36. responding to oligomers ranging from trimers up to 12mer8 (Table 55) were given as input to MACO 4 employing eqs 36 and 37. Three different chain models Z1,22,and 23, were considered: Model Z1 is a pure Bemoullian distribution defined by p(BU) = 0.65,p(ST) = 0.35,which gave the best agreement factor among pure Bernoullian distributions in the minimization process. Model 22 is a pure Markovian distribution having P matrix elements PBU,BU = 0.69,PBIJ,ST = 0.31,PST,BU = 0.39,and PST,ST = 0.61 with an associated composition p(BU) = 0.53,p(ST) = 0.47and ( n ~ u=)3.22, ST) = 1.64,which gave the best agreement factor among pure Markovian distributions in the minimization process. Model 23 is a pure Markovian distribution having P matrixelementsPBuau = 0.22,PBUsT = 0.78,PST,BU 0.90,and PsT,sT = 0.10with an associated

Iv

ysis MS, which results in thermally degrading the copolymer directly in the spectrometer.21 E1 mass spectra were recorded at temperatures below 480 O C . At these values, the chain cleavage process is totally selective, since PO units are stable and chain cleavage occurs exclusively at ET units21 (Figure 2a). Mass spectra of these copolyesters are quite complex21 since they display many mass series corresponding to PET units and to different copolymer fragments. Each mass series contains only a small number of peaks, and furthermore the E1 peak intensities may not reflect the actual oligomer abundances. We attempted to perform our simulation by using only one mass series (Tables VI1 and VIII). Despite this, significantstatistical analysis could be performed, because the selective character of the pyrolysis greatly reduces the number of oligomers produced in the partial degradation process. To proceed to the microstructure determination, the observed MS peak intensities corresponding to oligomers generated in the partial pyrolysis of two samples of copolyester IV (molar ratio OB/ET = 30/70and 60/40, respectively21)were given as input to MACO 4 using eqs 36



Table VI1 Experimental. and Calculatedbd Relative Amounts of the Partial Pyrolysis Products from an OB-ET Copolymer (ET = Ethylene Terephthalate, OB = pOxybenzoate)21 obsda oligomer

m/z

FAB-MS

(ET)2(0B) (ET)z(OB)z (ET)3(OB) (ET)z(OB)3 (E'M0B)z (ET)h(OB) (E'MOJ33 (ET)4(0Bh

312 432 504 552 624 696 744 816

30.8 9.9 46.9 0 2.5 8.6 0 1.2

AF'

calcdb

calcd'

calcdd

30.8 6.3 50.5 0.1

30.8 9.6 47.2 0.4 3.1 7.6 0.3 1.0

30.8

2.2

8.7 0.1 1.0 0.088

0.026

14.2

of partial selectivity. The starting implicit equation is

Ixxxx = f(R;xxx,e) (41) where f indicates a generic dependence. For a chain that follows Bernoullian statistics, three limiting cases have to be considered. In the case of totally-selective cleavages at the comonomer B, eq 41 must reduce to

42.6 0.9 4.1 6.1 0.4 0.9 0.119

Relative intensities of the ions from electron impact mass spectrum taken from Figure 3 in ref 21. * Computed using eq 36 with p(OB) = 0.2 and p(ET) = 0.8. Computed using eq 36 with p(0B) = 0.3 and p(ET) = 0.7. Computed using eq 36 with p(OB) = 0.4 and p(ET) = 0.6. e Agreement factor computed using formula 1.

*

Table VIII Experimental. and Calculatedkd Relative Amounts of the Partial Pyrolysis Products from an OB-ET Copolymer (ET = Ethylene Terephthalate, OB = pOxybenaoate)*l obsda oligomer

m/z

FAB-MS

(ET)z(OB)z (ET)z(OB) 03'h(OB)3 (ETMWz (ET)4(0B) (ET)dOB)3 (ET)AOB)z

432 504 552 624 696 744 816

27.0 39.3 4.1 9.8 4.1 7.4 8.2

AFe

calcdb

calcdC

calcdd

22.1 44.2 2.6 7.7 7.7 6.2 9.3

28.4 37.9 4.2 8.3 5.5 7.7 7.8

35.7 30.6 6.3 8.2 3.5 9.5 6.1

0.167

0.058

0.778

Relative intensities of the ions from the electron impact mass spectrum taken from Figure 4 in ref 21. Computed using eq 36 with p(OB) = 0.5 and p(ET) = 0.5. Computed using eq 36 with p(0B) = 0.6 and p(ET) = 0.4. Computed using eq 36 with p(0B) = 0.7 and p(ET) = 0.3. e Agreement factor computed using formula 1.

Z B ~= p(B)p(B)p(B)

(42)

In the case of totally-selective cleavages at comonomer A the explicit form of eq 41 is given by eq 36. Furthermore, eq 41 must reduce to eq 13 in the case of nonselective cleavages. We have found that these three constraints are sufficient to define a solution of eq 41, which is the following:

ZA = E€p(A)

and 37, valid for totally-selectiveprocesses. Both samples turned out to follow Bernoullian distributions, with bestfit values p(OB) = 0.30 and p(OB) = 0.60, respectively (Tables VI1 and VIII). The good agreement between the known2' and the computed composition confirms the validity of our statistical modeling. In the case of copolyester IV, the availability of MS data for four copolymer samples of different compositions2l allows us to test the predictions of eq 36. Figure 7 reports the normalized intensities measuredz1for each copolymer for the peak at m/z= 696 (correspondingto the (ET)d(OB) pentamer) versus the molar fraction of the E T component in thecopolymers. The experimental data21are compared in Figure 7 with the theoretical probability curve of the pentamer (ETMOB) applying eq 36 to the four copolymers, and it can be seen that the agreement between the two curves is a c ~ e p t a b l e .Remarkably, ~~ the probability curves generated by eq 36 (validfor the totally-selectiveprocesses) have the same shape as those generated by eq 13 (valid for nonselective cleavages) and they differ only by a proportionality factor. 9. Oligomer Distributions Generated by Partially-Selective Chain Cleavage Processes Equation 3 is an exact formula, but of very limited applicability because it describes the cleavage process making use of many variables, denoted generically as 0. An approximate solution of eq 3 can be derived on the assumption that the cleavage process can be described by a composition-independent variable, 6 , which is an index

= (1 -

- €)p(B)p(B)p(B)

(43)

Analogously, if a chain follows Markovian statistics, one

Macromolecules, Vol. 25,No.17,1992 obtains

Statistical Modeling of Mass Spectra of Copolymers 4279 A.F. 1

I

0.4

I&= t&IPAA

1 I

O0 2I

0

.- - :, ,

01

02

03

04

05

06

07

OS

08

1

INDEX OF PARTIAL SELECTIVITY ( 6 )

Figure 8. Agreement factor (AF) between experimental and

computed MS peak intensities as a function of the index of partial selectivity (c) in the case of copolyamide I11 (molar ratio A P / T P = 20180).

eq 43. The values of p(AP), p(TP), and c which yielded a minimum for the AF were p(AP) = 0.20, p(TP) = 0.80, and t = 1.0. Figure 8 reports the agreement factors between the observed and calculated MS peak intensities for the AP/TP copolyamide as a function of the index of partial selectivity t in the cleavage process. The plot, which displays the best AF for t = 1.0, was obtained keeping fixed the copolymer composition at values p(AP) = 0.20 and p(TP) = 0.80 in eq 43, and the result is the expected one, since the photolysis of copolyamide I11 is totally selective. In the case of a sequential disrtibution we have

IA = €&AIL

Acknowledgment. Partial financial support from the Italian Ministry for University and for Scientificand Technological Research (MURST) and from the National Council of Research (CNR) (Rome), Finalized Project of Fine and Secondary Chemistry,is gratefullyacknowledged. Supplementary Material Available: Experimental and calculated relative amounts of preformed MA-BA oligomers (Table lS), of the partial aminolysis products from a RE-BP copolycarbonate (Table 2.9, of t h e partial pyrolysis products from a n AN-BU copolymer (Table 3 9 , of the partial photolysis products of a n AP-TP copolyamide (Table 4S),and of the partial ozonolysis products from a BU-ST copolymer (Table 5s);Figure 1sshowing a plot of composition estimate vs oligomer groups (7 pages). Ordering information is given on any current masthead page.

References and Notes

Equations 43-45 constitute a complete group of equations that can be used to obtain the intensities of the MS peaks of Bernoullian, Markovian, and sequential copolymers in the case of partially-selective processes. Although examples of partially-selective cleavage processes are not yet available, it is possible at least to check the correctness of eq 43. For this purpose, the observed'' MS peak intensities for copolyamide I11 were given as input to MACO 4, using

(1) Montaudo, G. Rapid Commun. Mass Spectrosc. 1991,5,95 and references therein. Montaudo, G.; Puglisi, C. In Developments in Polymer Degradation; Grassie, N., Ed.; Elsevier: London, 1987;Vol. 7, p 35. Foti, S.; Montaudo, G. In Analysis ofPolymer Systems; Bark, L. S., Allen, N. S., Eds.; Elsevier: London, 1982; p 103. Montaudo, G. Br. Polym. J. 1986,18, 231. (2) Schulten, H.-R.; Lattimer, R. P. Mass Spectrom. Reu. 1984,3, 231 and references therein. (3) Barber, M.; Bordoli, R. S.; Elliot, G. J.; Sedgwick, R. D.; Tyler, A. N. Anal. Chem. l982,54,645A. Barber, M.; Bordoli, R. S.; Elliot, G. J.; Sedgwick, R. D.; Tyler, A. N. Nature 1981, 293, 270. (4) Burlingame, A. L.; Castagnoli, N., Jr., Eds. Mass Spectroscopy in the Health and Life Sciences; Elsevier: Amsterdam, 1986. (5) Bleetos, I. V.; Hercules, D. M.; Vanleyen, D.; Benninghoven, A.; Karakateanis, C. G.; Rieck, J. N. Anal. Chem. 1989,61,2142. (6) Lattimer, R. P.; Munster, H.; Budzikiewicz, H. Znt. J. Mass Spectrom. Zon Processes 1989,90, 119. (7) Garozzo, D.; Giuffrida, M.; Montaudo, G. Macromolecules 1986, 19,1643. Ballistreri, A.; Garozzo,D.; Giuffrida, M.; Montaudo, G. Anal. Chem. 1987,59,2024;Macromolecules 1986,19,2693. Ballistreri, A.; Garozzo, D.; Giuffrida,M.; Impallomeni,G.; Montaudo, G. Polym. Degrad. Stab. 1988,23, 25. Montaudo, G.; Puglisi, C.; Scamporrino, E.; Vitalini, D. Macromolecules 1986, 19, 2157. Montaudo, G.; Puglisi, C.; Samperi, F. Polym. De-

4280 Montaudo and Montaudo grad. Stab. 1989,26,285.Montaudo, G.; Puglisi, C.; Rapisardi, R.; Samperi, F. Polym. Degrad. Stab. 1991,31,229,291.Montaud0.G.: Scammrrino. E.: Vitalini. D. Makromol. Chem.. R a d Commun. 1989; 10,431. (8) . , Nuwavsir. L. M.: Wilkins. C. L.: Simonsick. W. J.. Jr. J . Am. SOC. Mass Specbom. 1990,1,66 and references therein. (9) Maravigna, P.; Montaudo, G. In Comprehensive Polymer Science; Pergamon Press: London, 1989;Vol. 5,p 63. Mandolini, L.; Montaudo, G.; Roelens, s.;Scamporrino, E.; Vitalini, D. Macromolecules 1989,22,3275.Montaudo, G. Macromolecules 1991,24,5829.Montaudo, G.; Puglisi, C.; Scamporrino, E.; Vitalini, D. Macromolecules 1988,21,1594.Ballistreri, A.; Garozzo,D.; Giuffrida, M.; Impallomeni, G.; Montaudo, G. Anal. Chem. 1990,62,279;Macromolecules 1987,20,1029. (io) Montaudo, M. S.;Ballistreri, A.;Montaudo, G. Macromolecules 1991,4,5051. (11) Ballistreri, A.; Garozzo, D.; Giuffrida, M.; Montaudo, G.; Montaudo, M. S. Macromolecules 1991,24, 1231. (12)Ballistreri, A.;Garozzo,D.; Giuffrida,M.; Impallomeni, G.; Montaudo, G. Macromolecules 1989,22,2107. Ballistreri, A.; Impallomeni, G.; Montaudo, G.; Lenz, R.; Kim, B.; Fuller, R. C. Macromolecules 1990,23,5059. (13) Randall, J. C. Polymer Sequence Determination; Academic Press: New York, 1977 and references therein. Bovey, F. A. Acc. Chem. Res. 1968,I, 175. Price, F. P. J.Chem. Phys. 1962, 36,209. Tonelli, A. E.NMR Spectroscopy and Polymer Microstructure; VCH Publishers: New York, 1989. (14)Kamiya, N.; Yamamoto, Y.; Inoue, Y.; Chujo, R.; Doi, Y. Macromolecules 1989,22, 1676. (15)Rempp, P.; Merrill, E. W. Polymer Synthesis: Huethig & Wepf Basel, 1991;p 108. (16)Plage, B.; Schulten, H.-R. Angew. Makromol. Chem. 1991,184, 133.

Macromolecules, Vol. 25,No.17,1992 (17) , , Montaudo. G.: Scamoorrino. . E.:, Vitalini. D. Macromolecules

1989,22,623.’ (18) Montaudo, G.; Puglisi, C.; Samperi, F. Polym. Bull. 1989,21, 483. (19)Montaudo, G.; Scamporrino, E.; Vitalini, D. Macromolecules 1989,22,627. (20)Montaudo, G.; Scamporrino, E.; Vitalini, D. Macromolecules 1991,24,376. (21)Garozzo, D.; Giuffrida, M.; Montaudo, G.; Lenz, R. W. J.Polym. Sci., Part A: Polym. Chem. 1987,25,271. (22) Source listing of program MACO 4 is available on request. (23) Knuth, D. E.The Art of Scientific Programming; AddisonWesley: Reading, MA, 1973;Vol. 3. (24)Harwood, J. Angew. Chem., Int. Ed. Engl. 1965,4,394. (25) Grassie, N.; Bain, D. R. J. Polym. Sci., Part A 1970,8, 2561, 2665. (26)Sometimes the composition can be calculated simply from the ratio IdIB when MS peak intensities corresponding to monomers A and B are reliable. Attempts to obtain approximate composition estimates have appeared in the most recent literature.8.2027 (27)Scamporrino, E.; Vitalini, D. Polymer, in press. (28)The above discussion has been restricted to the totally-selective cleavage processes involving only the cleavage of A units (Figure 2). Therefore, the selection rules derived here for the oligomers formed in the partial degradation process are valid strictly for this case. On the other hand, if cleavage of an alternating AB copolymer occurs at A-B linkages and not at B-A linkages, then only even oligomers will appear. However, examples of the latter type are not yet available in the literature. (29)The same order of agreement was found for the remaining oligomer groups (AzB, AzB2, A,B), whose normalized intensities are reported in Figure 6 of ref 21.

Further studies on the composition and microstructure of copolymers

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